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\begin{document}

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\section*{rigged limit}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory}

[[!include 2-category theory - contents]]

\hypertarget{rigged_limits}{}\section*{{Rigged limits}}\label{rigged_limits}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{characterization}{Characterization}\dotfill \pageref*{characterization} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{rigged limit} is a [[2-limit]] which is [[created limit|created]] in 2-categories of algebras and [[lax morphisms|lax]], colax, or pseudo morphisms for a 2-monad.

In order to characterize these most precisely, however, it turns out to be convenient to generalize from [[2-categories]] to [[F-categories]], using the corresponding notions of $\mathcal{F}$-monad, $\mathcal{F}$-limit, and so on.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $D$ be a [[small category|small]] strict $\mathcal{F}$-[[F-category|category]]. Then we have the [[functor category|functor]] $\mathcal{F}$-category $[D,\mathcal{F}]$ (where $\mathcal{F}$ denotes the $\mathcal{F}$-category $\mathcal{F}$). An object of $[D,\mathcal{F}]$ is an $\mathcal{F}$-functor $\Phi\colon D\to \mathcal{F}$, which can be identified with a pair of 2-functors $\Phi_\tau\colon D_\tau \to Cat$ and $\Phi_\lambda\colon D_\lambda\to Cat$ together with a 2-natural transformation

\begin{displaymath}
\itexarray{D_\tau & & \overset{J_D}{\to} & & D_\lambda\\
  & {}_{\Phi_\tau}\searrow & \neArrow & \swarrow_{\Phi_\lambda} \\
  & & Cat }
\end{displaymath}
whose components are full embeddings (objects of $\mathcal{F}$).

The [[tight morphisms]] in $[D,\mathcal{F}]$ are $\mathcal{F}$-natural transformations in the usual sense of enriched category theory, whereas its [[loose morphisms]] are 2-natural transformations between loose parts.

We also have an $\mathcal{F}$-category $Oplax(D,\mathcal{F})$ with the same objects, whose loose morphisms are [[oplax natural transformations]] between loose parts which are strictly 2-natural on tight morphisms, and whose tight morphisms are those whose components are all tight. The inclusion

\begin{displaymath}
[D,\mathcal{F}] \to Oplax(D,\mathcal{F})
\end{displaymath}
has a left adjoint, which induces an $\mathcal{F}$-[[comonad]] $\mathcal{Q}_c^D$ on $[D,\mathcal{F}]$.

\begin{udefn}
A weight $\Phi\colon D\to \mathcal{F}$ is \textbf{$l$-rigged} if

\begin{enumerate}%
\item It is a $\mathcal{Q}_c^D$-coalgebra, and
\item The canonical functor $Lan_{J_D} \Phi_\tau \to \Phi_\lambda$ is surjective on objects.

\end{enumerate}
\end{udefn}
We obtain definitions of \textbf{$c$-rigged} and \textbf{$p$-rigged} weights if we replace $Oplax(D,\mathcal{F})$ by $Lax(D,\mathcal{F})$ and $Pseudo(D,\mathcal{F})$, respectively.

\hypertarget{characterization}{}\subsection*{{Characterization}}\label{characterization}

Let $w$ denote one of $l$, $c$, or $p$.

\begin{utheorem}
For an $\mathcal{F}$-weight $\Phi$, the following are equivalent.

\begin{enumerate}%
\item $\Phi$ is $w$-rigged.
\item For any $\mathcal{F}$-monad $T$ on an $\mathcal{F}$-category $K$, the $\mathcal{F}$-functor $U_w\colon T Alg_w \to K$ creates $\Phi$-weighted limits.
\item For any 2-monad $T$ on a 2-category $K$, the functor $U_w\colon T Alg_w \to K$ (where $K$ denotes the chordate $\mathcal{F}$-category on $K$) creates $\Phi$-weighted limits.

\end{enumerate}
\end{utheorem}
See (\hyperlink{LS}{LS}) for the proof.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The following limits are $l$-rigged.

\begin{itemize}%
\item The [[2-limit]] of any diagram of tight morphisms which is also a limit as a diagram of loose morphisms. This includes any [[product]] and any [[power]].


\item The [[oplax limit]] of any diagram of loose morphisms.


\item The [[inserter]] of a [[parallel pair]] $f,g\colon A\to B$ such that $f$ (the domain of the 2-cell to be inserted) is tight. Here the projection to $A$ is tight and tightness-detecting.


\item The [[equifier]] of a parallel pair of 2-cells between a parallel pair of 1-morphisms $f,g\colon A\to B$ such that $f$ (the domain of the 2-cells) is tight. Again, the projection to $A$ is tight and tightness-detecting.


\item The [[Eilenberg-Moore object]] of a loose monad. Here the canonical forgetful morphism is tight and tightness-detecting.



\end{itemize}
Each has a fairly obvious dual version which is $c$-rigged. There are $p$-rigged versions as well, but $p$-rigged weights are almost equivalent to [[PIE-limits]]; see (\hyperlink{LS}{LS}) for details.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item Some limits of diagrams involving both lax and colax morphisms can also be given a $T$-algebra structure; see for instance [[colax/lax comma object]].

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Stephen Lack]], ``Limits for lax morphisms''. \emph{Appl. Categ. Structures}, 13(3):189--203, 2005


\item [[Stephen Lack]] and [[Mike Shulman]], ``Enhanced 2-categories and limits for lax morphisms'', \href{http://arxiv.org/abs/1104.2111}{arXiv}.



\end{itemize}
[[!redirects rigged limit]] [[!redirects rigged limits]] [[!redirects l-rigged limit]] [[!redirects l-rigged limits]] [[!redirects c-rigged limit]] [[!redirects c-rigged limits]] [[!redirects p-rigged limit]] [[!redirects p-rigged limits]] [[!redirects rigged weight]] [[!redirects rigged weights]]



\end{document}